-
Weak operator topology, operator rangesand operator equations
via Kolmogorov widths
M. I. Ostrovskii and V. S. Shulman
Abstract. Let K be an absolutely convex infinite-dimensional
compact in aBanach space X . The set of all bounded linear
operators T on X satisfyingTK ⊃ K is denoted by G(K). Our starting
point is the study of the closureWG(K) of G(K) in the weak operator
topology. We prove that WG(K)
contains the algebra of all operators leaving lin(K) invariant.
More preciseresults are obtained in terms of the Kolmogorov
n-widths of the compact K.The obtained results are used in the
study of operator ranges and operatorequations.
Mathematics Subject Classification (2000). Primary 47A05;
Secondary 41A46,47A30, 47A62.
Keywords. Banach space, bounded linear operator, Hilbert space,
Kolmogorovwidth, operator equation, operator range, strong operator
topology, weak op-erator topology.
1. Introduction
Let K be a subset in a Banach space X . We say (with some abuse
of the language)that an operator D ∈ L(X ) covers K, if DK ⊃ K. The
set of all operatorscovering K will be denoted by G(K). It is a
semigroup with a unit since theidentity operator is in G(K). It is
easy to check that if K is compact then G(K) isclosed in the norm
topology and, moreover, sequentially closed in the weak
operatortopology (WOT). It is somewhat surprising that for each
absolutely convex infinite-dimensional compact K the WOT-closure of
G(K) is much larger than G(K) itself,and in many cases it coincides
with the algebra L(X ) of all operators on X . Ouraim is to
understand: how much freedom has an operator which is obliged to
covera given compact? In a simplest form the question is: “How
large is G(K)?”. Weanswer this question describing the WOT-closure
WG(K) of G(K) as well as itsclosure in the ultra-weak topology (for
the case of Hilbert spaces). These resultsare obtained in Sections
2–3 for the Banach spaces, and in more detailed form
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2 M. I. Ostrovskii and V. S. Shulman
in Section 4 for Hilbert spaces; they are formulated in terms of
Kolmogorov’sn-widths of K.
In Section 5 we consider a more general object: the set G(K1,K2)
of alloperators T which have the property TK1 ⊃ K2 where K1,K2 are
fixed convexcompacts in Hilbert spaces.
In further sections we apply the obtained results for study of
some relatedsubjects: operator ranges (Section 7), operator
equations of the form XAY = B(Section 8) and operators with the
property
‖AXx‖ ≥ ‖Ax‖ for all x ∈ H
where A is a given operator on a Hilbert spaceH. Some
applications of the obtainedresults to the theory of quadratic
operator inequalities and operator fractionallinear relations will
be presented in a subsequent work. In fact our interest to
thesemigroups G(K) was initially motivated by these applications;
the relations toother topics became clear for us in the process of
the study.
Notation. Our terminology and notation of Banach space theory
follows [10]. Ourdefinitions of the standard topologies on spaces
of operators follow [3, ChapterVI]. Let X ,Y be Banach spaces. We
denote the closed unit ball of a Banach spaceY by BY , and the norm
closure of a set M ⊂ Y by M . We denote the set ofbounded linear
operators from Y to X by L(Y,X ). We write L(X ) for L(X ,X ).The
identity operator in L(X ) will be denoted by I.
Throughout the paper we denote by lin(K) the linear span of a
set K, andby VK the closed subspace spanned by K, that is, VK =
lin(K). We denote by AKthe algebra of all operators for which VK is
an invariant subspace. It is clear thatAK is closed in the WOT.
Remark on related work. Coverings of compacts by sets of the
form R(BZ) whereZ is a Banach space and R ∈ L(Z,X ) have been
studied by many authors, see[1], [2], and [6]. However, the main
foci of these papers are different. In all of thementioned papers
additional conditions are imposed on Z, or on R, or on bothof them,
and the main problem is: whether such R exist? In the context of
thepresent paper existence is immediate, while for us (as it was
mentioned above) themain question is: “How large is the set of such
operators?”.
We finish the introduction by showing that for non-convex
compacts K thesemigroup G(K) can be trivial:
Example. There exists a compact K in an infinite-dimensional
separable Hilbertspace H such that the only element of G(K) is the
identity operator.
Proof. Let {en}∞n=1 be an orthonormal basis in H; {αn}∞n=1 be a
sequence of realnumbers satisfying αn > 0 and limn→∞ αn+1/αn =
0; and {βn}∞n=1 be a sequenceof distinct numbers in the open
interval
(12 , 1). The compact K is defined by
K = {0} ∪ {αnen}∞n=1 ∪ {αnβnen}∞n=1 .
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Operator ranges 3
Assume that there exists D ∈ L(H) such that D(K) ⊃ K and D is
not theidentity operator. Let M = {n ∈ N : Den 6= en}. Since
{en}∞n=1 is a basis inH and D is not the identity operator, the set
M is nonempty. We introduce thefollowing oriented graph with the
vertex set M . There is an oriented edge −→nmstarting at n ∈M and
ending at m ∈M (n can be equal to m) if and only if oneof the
following equalities holds:
D(αmem) = αnen, D(βmαmem) = αnen,
D(αmem) = βnαnen, D(βmαmem) = βnαnen.(1)
Important observation. Since the numbers {βn}∞n=1 ⊂(
12 , 1)
are distinct, the num-ber of edges starting at n is at least 2
for each n ∈M , while there is at most oneedge ending at m ∈M .
An immediate consequence of this observation is that there are
infinitelymany oriented edges −→nm with n < m, that is,
infinitely many pairs (n,m), n < m,for which one of equalities
from (1) holds. Taking into account the conditionssatisfied by {αn}
and {βn}, we get a contradiction with the boundedness of D. �
This example shows that in the general case there is a very
strong dependenceof the size of the semigroup G(K) on the geometry
of K. To relax this dependencewe restrict our attention to
absolutely convex compacts K.
2. AK ⊂ WG(K)Theorem 2.1. Let K be an absolutely convex
infinite-dimensional compact. ThenAK ⊂WG(K).
Remark 2.2. Theorem 2.1 is no longer true for finite-dimensional
compacts. Infact, if K is absolutely convex finite-dimensional
compact, then A ∈ G(K) impliesthat A leaves VK invariant. Since VK
is finite-dimensional, the condition AK ⊃ Kpasses to operators from
the WOT-closure. Thus WG(K) is a proper subset of AK .
Let F be a subset of X ∗. We use the notation F⊥ for the
pre-annihilator ofF , that is, F⊥ := {x ∈ X : ∀f ∈ F f(x) = 0}.
Lemma 2.3. Let K be an absolutely convex infinite-dimensional
compact in a Ba-nach space X . For each finite-dimensional subspace
F ⊂ X ∗, each finite-dimen-sional subspace Y ⊂ X , and an arbitrary
linear mapping N : Y → X satisfyingN(Y ∩ VK) ⊂ lin(K), there is D ∈
G(K) satisfying the condition
Dx−Nx ∈ F⊥ ∀x ∈ Y. (2)
Proof. Note first of all that it suffices to prove the lemma
under an additionalassumption that Y ⊂ VK . Indeed, suppose that it
is done, then in the general casewe choose a complement Y1 of Y ∩
VK in Y and choose a complement X1 of Y1in X that contains VK . By
our assumption there is an operator D ∈ L(X1) with
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4 M. I. Ostrovskii and V. S. Shulman
DK ⊃ K and Dx − Nx ∈ F⊥ ∩ X1 ∀x ∈ Y ∩ VK . It remains to extend
D to Xsetting Dx = Nx for x ∈ Y1.
So we assume that Y ⊂ VK . For brevity denote lin(K) by Z.Let P
: X → X be a projection of finite rank, such that PX ⊃ Y + NY,
(I−P )X ⊂ F⊥ and dim(PZ) ≥ dimF +dimY. The last condition can be
satisfiedsince K is finite-dimensional.
The conditions N(Y ∩ VK) ⊂ lin(K) and Y ⊂ VK imply that the
subspaceNY is contained in Z ∩ PX . The space ker(P ) ∩ Z has
finite codimension in Z.Therefore there exists a complement L of
ker(P ) ∩ Z in Z such that L ⊃ NY.
We have PL = PZ and L∩(I−P )X = {0}. Since the subspace (I−P )X
hasfinite codimension in X , we can find a subspace M ⊃ L, which is
a complementof (I − P )X in X . Let QM : X → M be the projection
onto M with the kernelker(P ) = (I −P )X , and let M0 be the
complement of L in M . Since Z = (kerP ∩Z)⊕ L and QM (L) = L, we
have QM (Z) = L ⊂ Z.
To introduce an operator D ∈ L(X ) it suffices to determine its
action onkerP and on M . We do it in the following way:
(a) The restriction of D to kerP is a multiple λIkerP of the
identity operator,where λ is chosen in such a way that (I −QM )K ⊂
λ2K (such a choice is possiblebecause (I −QM )K is a compact subset
of Z = ∪n∈NnK).
(b) The restriction of D to M is defined in three ‘pieces’:
• D|M0 = 0.• Now we define the restriction of D to QM (Y).
Observe that QM (Y) ⊂ L.
This follows fromY ⊂ VK ⊂ L⊕ (I − P )X .
In addition QM |Y is an isomorphism, because Y ⊂ PX , and PX and
M arecomplements of the same subspace. Because of this, the
operator D|QM (Y)given by
D(QM (x)) = Nx+ αS(QM (x)) for x ∈ Y
is well-defined, where α ∈ R and S is an isomorphism of QM (Y)
into F⊥ ∩L.Such isomorphisms exist because dimL ≥ dim(PL) =
dim(PZ), and weassumed that dim(PZ) ≥ dimF + dimY. Now we choose α
to be so largethat the image of K ∩QM (Y) covers a ‘large’ multiple
of the intersection ofQM (K) with the space onto which it maps.
This is possible because zero hasnon-empty interior in K ∩QM (Y)
and QM (K) is compact.
• We define D on the complement of QM (Y) in L as a ‘dilation’
operator ontosome complement of the D(QM (Y)) in L. The number α
and the dilation areselected in such a way that
D(K ∩ L) ⊃ 2QM (K). (3)
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Operator ranges 5
To see that it is possible recall that QM (K) ⊂ L and, since L
is finite-dimensional,the set K ∩ L contains a multiple of the unit
ball of L.
It remains to verify that D satisfies the conditions (2) and DK
⊃ K.
Condition (2). Let x ∈ Y, then x = QMx+ (I −QM )x. Therefore
Dx = Nx+ αS(QM (x)) + λ(I −QM )x.
Let f ∈ F . We get:
f(Dx) = f(Nx) + αf(S(QM (x))) + λf(I −QM )x = f(Nx),
where we use the following facts: (a) The image of S is in F⊥;
(b) (I −QM )X =(I − P )X ⊂ F⊥.
Condition DK ⊃ K. Let x ∈ K. Then x = QMx+ (I−QM )x. The
condition(3) implies that there exists v ∈ 12 (L ∩ K) such that Dv
= QMx. The choice ofλ implies that w = 1λ (I −QM )x satisfies w
∈
12K. Let z = v + w. It is clear that
z ∈ K. We need to show that Dz = x. We have
Dz = Dv +Dw = Dv +D(
1λ
(I −QM )x)
= QMx+ λ(
1λ
(I −QM )x)
= x.
(We use the fact that (I −QM )X ⊂ kerP .) �
Proof of Theorem 2.1. Let T ∈ AK , and
U = {E ∈ L(X ) : ∀i ∈ {1, . . . , n} |fi(Exi)| < ε}
be a WOT-neighborhood of 0 in L(X ), where n ∈ N, ε > 0,
{fi}ni=1 ∈ X ∗ and{xi}ni=1 ∈ X . We need to show that T + U
contains an operator from G(K) foreach choice of n, ε, fi, and xi.
Let F = lin({fi}ni=1) and Y = lin({xi}ni=1). LetY1 = Y ∩ VK . Since
T ∈ AK , we have T (Y1) ⊂ VK . Since VK = Z, we can find a“slight
perturbation” T̃ of T satisfying T̃ (Y1) ⊂ Z. In particular, we can
find suchT̃ in T + 12U . It remains to show that T̃ +
12U contains an operator D from G(K).
It is clear that each operator S satisfying
∀x ∈ Y Sx− T̃ x ∈ F⊥
is in T̃ + 12U . Now the existence of the desired operator D is
an immediate conse-quence of Lemma 2.3 applied to N = T̃ . �
Corollary 2.4. If VK = X , then WG(K) = L(X ).
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6 M. I. Ostrovskii and V. S. Shulman
3. Application of Kolmogorov n-widths to estimates of the ‘size’
ofWG(K) from above
We are going to use the notion of Kolmogorov n-width. In this
respect we followthe terminology and notation of the book [17,
Chapter II]. Let Z be a subset of aBanach space X and x ∈ X . The
distance from x to Z is defined as
E(x,Z) = inf{||x− z|| : z ∈ Z}.
Definition 3.1. Let K be a subset of a Banach space X , n ∈ N ∪
{0}. The Kol-mogorov n-width of K is given by
dn(K) = infXn
supx∈K
E(x,Xn),
where the infimum is over all n-dimensional subspaces.
Lemma 3.2. Let K and K0 be two subsets in a Banach space X and D
∈ L(X ) besuch that D(K0) ⊃ K. Then dn(K) ≤ ||D||dn(K0) for all n ∈
N ∪ {0}.
Proof. Let Z ⊂ X be an n-dimensional subspace. Then DZ ⊂ X is a
subspace ofdimension ≤ n and E(Dx,DZ) ≤ ||D||E(x,Z). The conclusion
follows. �
Lemma 3.3. Let K be a bounded subset in a Banach space X . If K0
= K ∩ L,where L is a closed linear subspace in X which does not
contain K, then thereexists a constant 0 < C 12 for all M ≥ N
.Let 0 < ε < 1 and let xi ∈ K and scalars ai (i = 1, . . . ,
k) be such that thevector h =
∑i aixi satisfies ||h|| = 1 and ν(h) > 1 − ε. Let δ > 0 be
such that
δ||ν||∑|ai| < ε. Let N be such that for M ≥ N we have dM (K)
< δ/2. Then
for M ≥ N there exist yi ∈ LM such that ||xi − yi|| < δ.
Therefore the vectorg :=
∑i aiyi satisfies ||ν|| · ||g − h|| < ε and g ∈ LM . Choosing
appropriate ε and
δ we get ||ν|LM || > 12 .
Let M ≥ N and let LM,0 = LM ∩ ker ν. Let x ∈ K0. We are going to
showthat E(x, LM,0) < (2||ν|| + 1)2dM (K). By the definition of
LM there is y ∈ LMsuch that ||x−y|| ≤ 2dM (K). Since ν(x) = 0, we
have |ν(y)| ≤ ||ν||2dM (K). Since||ν|LM || > 12 , we conclude
that E(y, LM,0) < 4||ν||dM (K). Therefore E(x, LM,0)
<(2||ν||+ 1)2dM (K). It is clear that dimLM,0 = M − 1. Thus for
M ≥ N we havedM−1(K0) < (2||ν||+ 1)2dM (K). The conclusion
follows. �
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Operator ranges 7
Definition 3.4. Let {an} be a non-increasing sequence of
positive numbers satisfy-ing limn→∞ an = 0. We say that {an} is
lacunary if
lim infn→∞
an+1an
= 0. (4)
Lemma 3.5. If the sequence {dn(K)}∞n=1 is lacunary, then G(K) ⊂
AK .
Proof. Let R ∈ L(X ) be such that RVK is not contained in VK .
We have to showthat RK does not contain K. Assume the contrary.
It follows from our assumption that R−1(VK) is a proper subspace
of VK andR(K0) ⊃ K where K0 = K ∩R−1(VK) is a proper section of
K.
By Lemma 3.2 we get dn(K) ≤ ||R||dn(K0) for all n ∈ N ∪ {0}. By
Lemma3.3 we get dn(K0) ≤ Cdn+1(K) for some 0 < C < ∞ (which
depends on K andK0, but not on n) and all n ∈ N∪{0}. We get dn+1(K)
≥ (C||R||)−1dn(K), hencethe sequence {dn(K)}∞n=1 is not lacunary.
We get a contradiction. �
Remark 3.6. The assumptions of convexity and symmetry of K are
not needed inLemmas 3.2, 3.3, and 3.5.
Combining Theorem 2.1 and Lemma 3.5 we get
Theorem 3.7. If an absolutely convex compact K is such that the
sequence {dn(K)}is lacunary, then WG(K) = AK .
4. Covering of ellipsoids
4.1. s-numbers
Now we restrict our attention to the Hilbert space case, that
is, we consider sets Kof the form A(BH) where A is an
infinite-dimensional bounded compact operatorfrom a Hilbert space H
to a Hilbert space H1. Such sets are called ellipsoids.
Note. We continue using the Banach space theory notation and
terminology. Inparticular, unless explicitly stated otherwise, by
A∗ we mean the Banach-space-theoretical conjugate operator. It does
not seem that anything will be gained if weintroduce Hilbert-space
duality, but it can cause some confusion when we applyBanach space
case results for Hilbert spaces.
Remark 4.1. Many of the results below are true for A(BH) with
non-compact Aand usually the corresponding proofs are much simpler.
We restrict our attentionto the compact case.
Definition 4.2. (See [8, Chapter II, §2]) The eigenvalues of the
operator (E∗E)1/2(where E∗ is the conjugate in the Hilbert space
sense) are called the s-numbers ofthe operator E. Notation:
{sn(E)}∞n=1.
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8 M. I. Ostrovskii and V. S. Shulman
With this notation we have the following equalities for
n-widths:
dn(A(BH)) = sn+1(A)
(see [8, Theorem 2.2, p. 31]).
For ellipsoids we have a converse to the Lemma 3.2.
Lemma 4.3. If K0,K are ellipsoids in Hilbert spaces H1, H2,
respectively, anddn(K) ≤ Cdn(K0) for some C > 0 and all n ∈ N ∪
{0}, then there is an operatorD ∈ L(H1,H2) such that DK0 ⊃ K and
‖D‖ ≤ C.
Proof. The result follows from the so-called Schmidt expansion
of a compact op-erator (see [8, p. 28]), which implies that
K = A(BH) =
{ ∞∑n=1
αnsn(A)hn : {αn}∞n=1 ∈ `2,
{hn}∞n=1 is an orthonormal sequence}
and
K0 = B(BH) =
{ ∞∑n=1
αnsn(B)gn : {αn}∞n=1 ∈ `2,
{gn}∞n=1 is an orthonormal sequence} .
It is easy to see that there is a bounded linear operator D
which maps gn ontoChn, and that this operator satisfies the
conditions D(K0) ⊃ K, ‖D‖ ≤ C. �
Remark 4.4. The proof of Lemma 4.3 shows that the desired
operator D can beconstructed as an operator whose restriction to
VK0 is a multiple of a suitablechosen bijective isometry between
VK0 and VK , extended to H1 in an arbitraryway.
Known results on s-numbers imply the following lemma.
Lemma 4.5. Let K be an ellipsoid in a Hilbert space H such that
{dn(K)}∞n=0 isnot lacunary. Let K0 be the intersection of K with a
closed linear subspace of finitecodimension. Then there exists δ
> 0 such that dn(K0) ≥ δdn(K) and a boundedlinear operator Q :
lin(K0)→ lin(K) satisfying Q(K0) ⊃ K.
Proof. LetA : H → H be a compact operator satisfyingK = A(BH).
The sequence{dn(K0)}∞n=0 is the sequence of s-numbers of a
restriction of A to a subspace offinite codimension. This sequence
is, in turn, the sequence of s-numbers of anoperator of the form
A+G, where G is an operator of finite rank.
It is known [8, Corollary 2.1, p. 29] that sn(A + G) ≥ sn+r(A),
where r isthe rank of G. Combining this inequality with the
assumption that the sequence{sn(A)}∞n=1 is not lacunary, we get the
desired inequality.
The last statement of the lemma follows from Lemma 4.3. �
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Operator ranges 9
4.2. WOTTheorem 4.6. If H is a Hilbert space, K ⊂ H is an
ellipsoid and the sequence{dn(K)} is not lacunary, then WG(K) =
L(H).
The definition of WOT shows that to prove Theorem 4.6 it
suffices to provethe following lemma.
Lemma 4.7. Let K be an ellipsoid in a Hilbert space H. Suppose
that the sequence{dn(K)} is non-lacunary. Then for each
finite-dimensional subspace Y ⊂ H andeach linear mapping N : Y → H,
there is an operator D satisfying conditions:Dy = Ny for all y ∈ Y,
and DK ⊃ K.Proof. Let Z = Y⊥ and K0 = K ∩Z. By Lemma 4.5 there is
an operator E fromZ to H with EK0 ⊃ K. Extend it to an operator D :
H → H setting Dy = Nyon Y. �
Remark 4.8. One can see from the proof of Lemma 4.7 that under
the statedconditions the closure of G(K) in the strong operator
topology coincides with L(H).Corollary 4.9. Let K be an ellipsoid
in a Hilbert space H. Then:(1) WG(K) = AK if the sequence
{dn(K)}∞n=0 is lacunary.(2) WG(K) = L(H) if the sequence
{dn(K)}∞n=0 is not lacunary.
4.3. Ultra-weak topology
It turns out that Theorem 4.6 remains true if we replace closure
in the weak opera-tor topology, by a closure in a stronger
topology, usually called ultra-weak topology.This topology on L(H)
is defined as the weak∗ topology corresponding to the du-ality L(H)
= (C1(H))∗, where C1(H) is the space of nuclear operators.
(Necessarybackground can be found in [18, Chapter II],
unfortunately the terminology andnotation there is different, the
ultra-weak topology is called σ-weak topology, see[18, p. 67]).
Ultra-weak and strong operator topologies are incomparable, for
thisreason our next result does not follow from Remark 4.8.
Theorem 4.10. If K is an ellipsoid in a Hilbert space H and the
sequence {dn(K)}is not lacunary, then the ultra-weak closure of
G(K) coincides with L(H).Proof. Let {Ti}mi=1 be a finite collection
of operators in C1(H) and R ∈ L(H). Itsuffices to show that there
is D ∈ L(H) satisfying
tr(DTi) = tr(RTi) for i = 1, . . . ,m and DK ⊃ K. (5)It is clear
that we may assume that the operators Ti are linearly
independent.
Lemma 4.11. If {Ti}mi=1 are linearly independent, then there
exists a finite rankprojection P ∈ L(H) such that the mapping
ω : L(H)→ Rm
given byω(U) = {tr(UPTi)}mi=1
is surjective.
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10 M. I. Ostrovskii and V. S. Shulman
Proof. We have to prove that there is a finite rank projection P
such that the oper-ators PTi are linearly independent (in this case
the mapping ω will be surjective).Using induction we may suppose
that P0T1, ..., P0Tm−1 are linearly independentfor some P0.
Consider the set M0 of those finite rank projections P which
commutewith P0 and satisfy imP ⊃ imP0. We claim that there exists P
∈ M0 such thatPT1, . . . , PTm are linearly independent.
Assume contrary, then for each P ∈M0, one can find λ1(P ), ...,
λm−1(P ) ∈ Csatisfying PTm =
∑m−1k=1 λk(P )PTk (using the definition of M0 it is easy to get
a
contradiction if PT1, . . . , PTm−1 are linearly dependent). Our
next step is to showthat the numbers {λk(P )}m−1k=1 do not depend
on P . In fact, for any P1, P2 ∈M0 wehave
∑k
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Operator ranges 11
Using this inclusion and elementary properties of WOT we get
WG(K2)WG(K1,K2)WG(K1) ⊂WG(K1,K2). (8)Lemmas 3.2 and 4.3 imply
that the set G(K1,K2) is non-empty if and only
ifdn(K2) = O(dn(K1)). (9)
From now on till the end of this section we assume that (9) is
satisfied.
Observation 5.1. By Remark 4.4, condition (9) implies that there
is an onto isom-etry M : VK1 → VK2 and a number α ∈ R+ such that
αM(K1) ⊃ K2. Con-sider decompositions H1 = VK1 ⊕ R1 and H2 = VK2 ⊕
R2. Let A1 ∈ L(VK1),B1 ∈ L(R1,H1), A2 ∈ L(VK2), B2 ∈ L(R2,H2), and
C : R1 → H2. CombiningTheorem 2.1 with (8) we get that the
composition (A2 ⊕ B2)(αM ⊕ C)(A1 ⊕ B1)is in WG(K1,K2), where Ai ⊕Bi
: VKi ⊕Ri → Hi, i = 1, 2.
To state our results on the description of WG(K1,K2) we need the
followingdefinitions.
Definition 5.2. The kth left shift of a sequence {an}∞n=0 (k ≥
0) is the sequence{an+k}∞n=0.
Definition 5.3. Let {an}∞n=0 and {bn}∞n=0 be sequences of
non-negative numbers.We say that {an}∞n=0 majorizes {bn}∞n=0 if
there is 0 < C
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12 M. I. Ostrovskii and V. S. Shulman
(B) Suppose that the kth left shift of {dn(K1)} majorizes
{dn(K2)}. Let T ∈L(H1,H2) be such that the dimension of the image
of the space T (VK1) in thequotient space H2/VK2 is ≤ k. We show
that T ∈WG(K1,K2).
Let F be a finite subset of H∗2 and Y be a finite subset of H1.
It suffices toshow that there exists D ∈ G(K1,K2) satisfying f(Dy)
= f(Ty) for each y ∈ Yand each f ∈ F . With this in mind, we may
assume that F and Y are finite-dimensional subspaces. Also, we may
assume that Y is a subspace of VK1 , becausewe may let the
restriction of D to the orthogonal complement of VK1 be the sameas
the restriction of T .
We decompose F as FO ⊕ FV , where FO = F ∩ V ⊥K2 . It is easy to
check thatthe assumption on T implies that (T ∗FO)⊥ ∩ VK1 is of
codimension at most k(if it is of codimension ≥ k + 1, then we can
find k + 1 vectors xi ∈ VK1 andk + 1 functionals x∗j in FO such
that T
∗x∗j (xi) = δi,j , but then x∗j (Txi) = δij
shows that {Txi} is a family of k+ 1 vectors whose images in
H2/VK2 are linearlyindependent, contrary to our assumption).
Now we decompose Y = Y1 ⊕ Y2, where Y1 = Y ∩ (T ∗FO)⊥. We let
D|Y2 =T |Y2 . Our next step is to find a suitable definition of the
restriction of D to(T ∗FO)⊥ ∩ VK1 . To this end we need the
following modification of Lemma 4.5,which can be proved using the
same argument and Remark 4.4.
Lemma 5.5. Let K1 and K2 be ellipsoids in Hilbert spaces H1 and
H2, respectively.Suppose that the kth left shift of {dn(K1)}∞n=0
majorizes {dn(K2)}∞n=0 and thatK0 is the intersection of K1 with a
subspace of H1 of codimension k. Then thereexists an operator B :
VK0 → VK2 such that B(K0) ⊃ K2 and B is a multiple of abijective
linear isometry of VK0 and VK2 .
Applying Lemma 5.5 we find an operator B : ((T ∗FO)⊥ ∩VK1)→ VK2
whichsatisfies B((T ∗FO)⊥ ∩K1) ⊃ K2 and is a multiple of a
bijective isometry. Now wemodify B using Lemma 2.3, which we apply
for X = VK2 , K = K2, Y = BY1,N = TB−1|BY1 , and F (which is
denoted in the same way in this proof). Wedenote the operator
obtained as a result of the application of Lemma 2.3 by H.
We let D|(T∗FO)⊥∩VK1 = HB. This formula defines D on Y1, and
this defini-tion is such that D|Y1 = T |Y1 . We extend D to the
rest of the space H1 arbitrarily.
It is clear that D satisfies all the assumptions. Thus T
∈WG(K1,K2).Now we suppose that the (k + 1)th left shift of {dn(K1)}
does not majorize
{dn(K2)} and show that if T is an operator for which T (VK1)
contains k + 1vectors whose images in the quotient space H2/VK2 are
linearly independent, thenT /∈WG(K1,K2).
Using the standard argument we find ε > 0, v1, . . . , vk+1 ∈
VK1 , and function-als f1, . . . , fk+1 ∈ H∗2, such that any D ∈
L(H1,H2) satisfying |fj(Dvi−Tvi)| < ε,i, j = 1, . . . , k + 1,
satisfies the condition: D(VK1) contains k + 1 vectors whose
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Operator ranges 13
images in H2/VK2 are linearly independent. It remains to show
that such operatorsD cannot satisfy DK1 ⊃ K2.
In fact the condition about k + 1 linearly independent vectors
implies thatD−1(VK2) ∩ VK1 is a subspace of VK1 of codimension at
least k + 1.
Therefore K2 is covered by a section K0 of K1 of codimension k+
1. On theother hand, by Lemma 3.3, the sequence of n-widths of K0
is majorized by the(k + 1)th left shift of {dn(K1)}∞n=1. By Lemma
3.2, we get a contradiction withour assumption. �
Corollary 5.6. If {dn(K1)}∞n=0 is non-lacunary and the condition
(9) is satisfied,then WG(K1,K2) = L(H1,H2).
In fact, if {dn(K1)}∞n=0 is non-lacunary, it is majorized by
each of its leftshifts, and hence the assumption of Theorem 5.4(A)
is satisfied.
Remark 5.7. In the case where {dn(K1)}∞n=0 is lacunary both the
situation de-scribed in Theorem 5.4(A) and the situation described
in Theorem 5.4(B) can oc-cur.
Similarly to the case of one compact we introduce
AK1,K2 := {T ∈ L(H1,H2) : TVK1 ⊂ VK2}.
The following is a special case of Theorem 5.4(B) corresponding
to the case k = 0:
Corollary 5.8. Let K1,K2 be ellipsoids with
lim infdn+1(K1)dn(K2)
= 0. (10)
Then WG(K1,K2) = AK1,K2 .
Remark 5.9. Note that the combination of the assumption (9) and
the condition(10) imply that the sequences {dn(K1)}∞n=1 and
{dn(K2)}∞n=1 are both lacunary.Indeed, dk(K2) ≤ Cdk(K1) implies
dn+1(K1)dn(K2)
≥ dn+1(K1)Cdn(K1)
anddn+1(K1)dn(K2)
≥ dn+1(K2)Cdn(K2)
.
Therefore (10) implies that {dn(K1)}∞n=1 and {dn(K2)}∞n=1 are
lacunary.
Analysis of all possible cases in Theorem 5.4 implies also the
following:
Corollary 5.10. If K1 and K2 are ellipsoids for which VKi = Hi
for i = 1, 2, and(9) is satisfied, then WG(K1,K2) = L(H1,H2).
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14 M. I. Ostrovskii and V. S. Shulman
6. Covering with compact operators
Here we discuss the problem of covering an ellipsoid K2 by the
image of an ellipsoidK1 via a compact operator. Let CG(K1,K2) be
the set of all compact operatorsT satisfying the condition TK1 ⊃
K2.
Let us begin with an analogue of Lemma 3.2.Note that the widths
dn(K) of a compact subset K in a Banach space X can
change if we consider K as a subset of a subspace Y ⊂ X that
contains K. Letus denote by d̃n(K) the n-width of K considered as a
subset of VK (recall thatVK = linK, so we choose the minimal
subspace and obtain maximal widths).
Lemma 6.1. Let X and Y be Banach spaces, K be a compact set in X
and T :X → Y be a compact operator. Then d̃n(TK)/d̃n(K)→ 0 as
n→∞.
Proof. We may assume that X = VK . By the definition of d̃n, for
each n ∈ N∪{0}and 0 < ε < 1, there exists an n-dimensional
subspace Xn ⊂ X such that
K ⊂ Xn + dn(K)(1 + ε)BX . (11)
ThereforeTK ⊂ TXn + dn(K)(1 + ε)TBX . (12)
Now we show that for each δ > 0 there is N ∈ N such that
TBX ⊂ TXn + δBY for n ≥ N. (13)
In fact, since TBX is compact, it has a finite δ/3-net {yi}ti=1
⊂ TBX . SinceTBX ⊂ TVK , the vectors yi can be arbitrarily well
approximated by linear combi-nations of vectors from TK. Let M be
the maximum absolute sum of coefficientsof a selection of such
δ/3-approximating linear combinations. Let N be such thatfor n ≥ N
we have dn(K) ≤ δ6M ||T || , and let us show that (13) holds.
We need to show that for all y ∈ TBX we have dist(y, TXn) ≤
δ.Let j ∈ {1, . . . , t} be such that ||y − yj || < δ/3, and
let
∑si=1 αiTxi be such
that xi ∈ K,∑si=1 |αi| ≤ M , and ||yj −
∑si=1 αiTxi|| < δ/3 . By (11), we have
dist(xi,Xn) ≤ dn(K)(1 + ε). Therefore
dist(s∑i=1
αiTxi, TXn) ≤s∑i=1
|αi|||T ||dn(K)(1 + ε) ≤M ||T || ·δ
6M ||T ||· (1 + ε) < δ
3.
Thus dist(y, TXn) < δ.If we combine (12) and (13) we get
dn(TK) ≤ (1 + ε)δdn(K) for n ≥ N .
Since 0 < ε < 1 and δ > 0 can be chosen arbitrarily,
the statement follows. �
Remark 6.2. Note that if X is a Hilbert space, then d̃n(K) =
dn(K). Indeed, in thiscase we may assume that Xn ⊂ VK . Such
subspace can be found as the orthogonalprojection to VK of any
subspace Xn satisfying (11). It should be mentioned thatin the case
where both X and Y are Hilbert spaces a simpler proof of Lemma
6.1is known, see [7, Lemma 1].
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Operator ranges 15
Now we find criteria of non-emptiness of CG(K1,K2) for
ellipsoids K1 andK2. The result can be considered as an analogue of
Lemmas 3.2 and 4.3.
Lemma 6.3. Let K1 and K2 be ellipsoids in Hilbert spaces H1 and
H2, respectively.There is a compact operator T satisfying TK1 ⊃ K2
if and only if
dn(K2) = o(dn(K1)). (14)
Proof. If there is a compact operator T with TK1 ⊃ K2 then (14)
follows fromLemma 6.1 and Remark 6.2. Conversely, if (14) holds,
then the existence of acompact operator T follows from the argument
of Lemma 4.3. �
If the condition (14) is satisfied we say: the sequence
{dn(K1)}∞n=1 strictlymajorizes {dn(K2)}∞n=1.
Let us define by WCG(K1,K2) the WOT-closure of CG(K1,K2). We
havethe following analogue of Theorem 5.4:
Theorem 6.4. (A) If all left shifts of the sequence {dn(K1)}
strictly majorize thesequence {dn(K2)}, then WCG(K1,K2) =
L(H1,H2).(B) If the kth left shift of {dn(K1)} strictly majorizes
the sequence {dn(K2)}, butthe (k+1)th left shift does not (such
cases are clearly possible), then WCG(K1,K2)is the set of those
operators T ∈ L(H1,H2) for which the image of the space T (VK1)in
the quotient space H2/VK2 is at most k-dimensional.
The proof is a straightforward modification of the proof of
Theorem 5.4 andwe omit it.
7. Operator ranges
In this section by a Hilbert space we mean a separable
infinite-dimensional Hilbertspace. An operator range is the image
of a Hilbert space H1 under a bounded op-erator A : H1 → H2.
Operator ranges are actively studied, see [2], [4], [9], [13],
andreferences therein. The purpose of this section is to use the
results of the previoussection to classify operator ranges. Our
results complement the classification ofoperator ranges presented
in [4, Section 2].
We restrict our attention to images of compact operators of
infinite rank. Theset A(BH1) will be called a generating ellipsoid
of the operator range AH1. Thesame operator range is the image of
infinitely many different operators, thereforea generating
ellipsoid of an operator range is not uniquely determined.
However,the Baire category theorem implies that if K1 and K2 are
generating ellipsoids ofthe same operator range, then cK1 ⊂ K2 ⊂
CK1 for some 0 < c ≤ C
-
16 M. I. Ostrovskii and V. S. Shulman
It is clear that a sequence is lacunary if and only if all of
sequences equivalentto it are lacunary. It is also clear that left
shifts of equivalence classes of sequencesare well-defined as well
as the conditions like d(Y1) majorizes d(Y2). Therefore
thefollowing notions are well-defined for operator ranges: (i) Y is
lacunary; (ii) Y1majorizes Y2. We say that an operator range Y ⊂ H
is dense if Y = H.
Results of Section 5 on covering of one ellipsoid by another
have immediatecorollaries for operator ranges. Let A1 : H → H1 and
A2 : H → H2 be compactoperators of infinite rank and Yi = AiH. Let
R(Y1,Y2) denote the set of alloperators T satisfying
TY1 ⊃ Y2. (15)We write R(Y) instead of R(Y,Y). The WOT-closure
of R(Y1,Y2) will be denotedby WR(Y1,Y2).
Corollary 7.1. Suppose that Y1 majorizes Y2. Then(i) If all left
shifts of d(Y1) majorize d(Y2), then WR(Y1,Y2) = L(H1,H2).
(ii) Let k be a non-negative integer. If the kth left shift of
d(Y1) majorizes d(Y2),but the (k + 1)th left shift does not, then
WR(Y1,Y2) is the set of thoseoperators T for which the image of TY1
in the quotient space H2/Y2 hasdimension ≤ k. In particular, if k =
0, we get: if the first left shift of d(Y1)does not majorize d(Y2),
then WR(Y1,Y2)Y1 ⊂ Y2.
(iii) If Y1 is non-lacunary, then WR(Y1,Y2) = L(H1,H2).(iv) If
Y1 and Y2 are dense, then WR(Y1,Y2) = L(H1,H2).
Proof. To derive (i)-(iv) from Theorem 5.4 and its corollaries
we need two obser-vations:• R(Y1,Y2) contains G(K1,K2) for any pair
of generating ellipsoids.• If T ∈ R(Y1,Y2) then TK1 ⊃ K2 for some
pair of generating ellipsoids.
The first observation immediately implies (i), (iii), (iv), and
“estimates frombelow” in (iv). The second observation shows that
for “estimates from above” in(ii) we can use the same argument as
in Section 5. �
One of the systematically studied objects in the theory of
invariant subspaces,see [5, 14, 15, 16], is the algebra A(Y) of all
operators that preserve invariant agiven operator range Y. It is
known, see [16, Theorem 1], that if Y is dense, thenthe WOT-closure
WA(Y) of A(Y) coincides with L(H). It follows easily that ingeneral
WA(Y) consists of all operators that preserve the closure Y of
Y.
An operator algebra A is called full if it contains the inverses
of all invertibleoperators in A. We call A weakly full if for each
invertible operator T ∈ A, theoperator T−1 belongs to the
WOT-closure of A. Our next result shows that foralgebras of the
form A(Y) this property depends on d(Y).
Corollary 7.2. (i) If the closure Y of an operator range Y ⊂ H
has finite codi-mension in H, then the algebra A(Y) is weakly
full.
(ii) If Y is not lacunary and codim(Y) =∞, then A(Y) is not
weakly full.
-
Operator ranges 17
(iii) If Y is lacunary, then A(Y) is weakly full.
Proof. (i) If T preserves Y then TY ⊂ Y. If T is invertible,
then it maps a com-plement of Y onto a complement of T (Y). If Y
has finite codimension, this impliesTY = Y. Hence T−1Y = Y, and T−1
is in the WOT-closure of A(Y).
(ii) Let K be a generating ellipsoid of Y. Choose a nonzero
vector y ∈ Y and letK0 = K∩y⊥. By Lemma 4.5, the sequences {dn(K)}
and {dn(K0)} are equivalent.Using Observation 5.1 we find an
operator D : VK0 → VK which satisfies D(K0) ⊃K and is a (nonzero)
multiple of an isometry. Since Y has infinite codimension, wecan
extend D to an invertible operator D : H → H. Observe that D(Y ∩
y⊥) = Y,therefore D(y) /∈ Y, and thus D /∈ WA(Y). On the other
hand, the inclusionD(K0) ⊃ K implies D−1 ∈ A(Y).
(iii) If T ∈ A(Y) is invertible, then T−1 ∈ R(Y). Since Y is
lacunary, applyingCorollary 7.1 we conclude that T−1 preserves Y.
Therefore T−1 ∈ WA(Y). �
8. Bilinear operator equations
One of the popular topics in operator theory is the study of
linear operator equa-tions XA = B and AX = B. We consider here a
“bilinear operator equation”
XAY = B, (16)
where operators A,B are given. Its solution is a pair (X,Y ) of
operators. We denotethe set of all such solutions by S(A,B). For
simplicity we restrict our attention tothe case when all operators
act on a fixed separable Hilbert space H. Such a pair(X,Y ) can be
found if we fix one of the operators (say X) and solve the
obtainedlinear equation (which has more than one solution in the
degenerate cases only). Sothe study of the question “how many
solutions does equation (16) have?” reducesto the study of the set
of all first components, that is, the set of those X for which(X,Y
) is a solution for some Y . Let us denote this set by U(A,B).
Corollary 8.1. (i) The equation is solvable if and only if
sn(B) = O(sn(A)). (17)
(ii) Suppose that condition (17) holds. If operators A,B have
dense ranges,or if the range of A is non-lacunary, then U(A,B) is
WOT-dense in L(H).
(iii) If the range of operator B is not dense and the
condition
sn(B) = O(sn+1(A)) (18)
does not hold, then U(A,B) is not WOT-dense in L(H).
Proof. Clearly X ∈ U(A,B) if and only if the equation (16) is
solvable with respectto Y . This is equivalent to the inclusion XAH
⊃ BH. It remains to apply Corollary7.1. �
-
18 M. I. Ostrovskii and V. S. Shulman
If an operator A is not compact then the set is WOT-dense in
L(H). Formallythis is not a special case of Corollary 8.1(ii)
because s-numbers are usually definedfor compact operators only,
but the proof in this case along the same lines is evensimpler. In
the rest of the section we prove that this result can be
considerablystrengthened: if A is not compact then S(A,B) itself is
dense in L(H)×L(H) withrespect to the weak (and even strong)
operator topology.
Lemma 8.2. For any two linearly independent families (x1, ...,
xn), (y1, ..., ym) ofvectors in H, two arbitrary families (x′1,
..., x′n), (y′1, ..., y′m) of vectors in H, and anumber � > 0,
there is an invertible operator V with the properties ‖V xi−x′i‖
< �,‖V −1yj − y′j‖ < � .
Proof. One can choose systems z1, ..., zn and w1, ..., wm close
to (x′1, ..., x′n) and,
respectively, (y′1, ..., y′m) in such a way that both
systems
(x1, ..., xn, w1, ..., wm) and (y1, ..., ym, z1, ..., zn)
are linearly independent. Let us define an operator T between
their linear spans byTxi = zi, Twj = yj . It is injective and
therefore can be extended to an invertibleoperator on a
finite-dimensional space containing these systems. Clearly an
invert-ible operator on a finite-dimensional subspace can be
extended to an invertibleoperator on the whole space (take the
direct sum with the identity operator). �
We denote the group of all invertible operators on H by G(H).
Note. In thissection A∗ denotes the Hilbert space conjugate of an
operator A.
Lemma 8.3. If an operator X has dense image and an operator Y
has trivial kernel,then the set
ΓX,Y = {(XV −1, V Y ) : V ∈ G(H)}is dense in L(H)× L(H) with
respect to the strong operator topology (SOT).
Proof. Let a system (x1, ..., xn), (y1, ..., ym), (x′1, ...,
x′n), (y
′1, ..., y
′m) and � > 0 be
given as above. The system x̃i = Y xi is linearly independent
since kerY = 0.Since XH is dense, there are zj with ‖Xzj − y′j‖
< �/2. Take 0 < δ < �2||X||and choose an invertible
operator V as in Lemma 8.2 for the system (x̃1, ..., x̃n),(y1, ...,
ym), (x′1, ..., x
′n), (z1, ..., zm) and δ. The obtained inequalities imply
that
ΓX,Y is SOT-dense in L(H)× L(H). �
Any solution (X,Y ) of the equation XY = B will be called a
factorizationof an operator B.
Proposition 8.4. For each operator B in an infinite-dimensional
Hilbert space H,the set P(B) of all its factorizations is SOT-dense
in L(H)× L(H).
Proof. Let H = H1 ⊕H2 where H1 and H2 are of the same dimension
as H. LetU1 and U2 be isometries with the ranges H1 and H2,
respectively. Then U∗1 andU∗2 isometrically map H1 and H2,
respectively, onto H, also U∗1H2 = {0} andU∗2H1 = {0}. We set Y =
U1 and X = BU∗1 + U∗2 .
-
Operator ranges 19
Since XY = BU∗1U1 + U∗2U1 = B, we have (X,Y ) ∈ P(B), and
therefore
(XV −1, V Y ) ∈ P(B) for each V ∈ G(H). It follows easily from
the definition ofoperators X,Y that XH = H and ker(Y ) = 0.
Applying Lemma 8.3 we concludethat P(B) is SOT-dense in L(H)× L(H).
�
Let us write A � B if the set S(A,B) of all solutions of (16) is
SOT-densein L(H) × L(H). For brevity, we will denote by Es the
closure of a subset E ofL(H)× L(H) with respect to the product of
SOT-topologies.
Lemma 8.5. If A � B and B � C, then A � C.
Proof. If (X,Y ) ∈ S(A,B) and (X1, Y1) ∈ S(B,C), then (XX1, Y1Y
) ∈ S(A,C).Taking (X1, Y1) → (I, I) we get that (X,Y ) ∈ S(A,C)
s. Hence L(H) × L(H) ⊂
S(A,C)s
and A � C. �
We proved in Proposition 8.4 that I � C for all C. So our aim is
to showthat A � I for each non-compact A.
Lemma 8.6. If P is a projection of infinite rank, then P �
I.
Proof. Let U be an isometry with UU∗ = P . Then (U∗, U) ∈ S(P,
I). Hence
(V U∗, UV −1) ∈ S(P, I) for each V ∈ G(H).
It follows that S(P, I)s
contains all pairs (M,N) with NH ⊂ PH, M(I − P ) = 0.Hence for
each (X,Y ) ∈ L(H)×L(H), the pair (XP,PY ) belongs to S(P, I)
s.
Choose a net (Xλ, Yλ) in S(P, I) with (Xλ, Yλ)→ (XP,PY ) in SOT,
then (Xλ +X(I −P ), Yλ + (I −P )Y ) ∈ S(P, I) (indeed (Xλ +X(I −P
))P (Yλ + (I −P )Y ) =XλPYλ = 1). Since (Xλ + X(I − P ), Yλ + (I −
P )Y ) → (X,Y ) we get that(X,Y ) ∈ S(P, I)
s. �
The proof of the next lemma is immediate.
Lemma 8.7. (i) If (X,Y ) ∈ S(F1AF2, I), then (XF1, F2Y ) ∈ S(A,
I).In particular(ii) If F1AF2 � I, ker(F1) = 0 and F2H = H then A �
I.
Lemma 8.8. Let A = 0 ⊕ A1, where A1 acts on infinite-dimensional
space and isinvertible. Then A � I.
Proof. Let F = I ⊕ A−11 then F is invertible and FA is a
projection of infiniterank. Hence FA � 1, by Lemma 8.6. Using Lemma
8.7 (ii), we get that A � 1. �
Lemma 8.9. If A ≥ 0 and A is not compact, then A � I.
Proof. For each ε > 0, let Pε = I − Q, where Q is the
spectral projection of Acorresponding to the interval (0, ε). Then
PεA is of the form 0 ⊕ B, where B isinvertible and, for
sufficiently small ε, non-compact. Hence PεA � I. By Lemma8.7,
L(H)Pε×L(H) ⊂ S(A, I)
s. Since Pε → I when ε→ 0, we get that A � I. �
-
20 M. I. Ostrovskii and V. S. Shulman
Theorem 8.10. If A is non-compact, then the set of all solutions
of the equation(16) is SOT-dense in L(H)× L(H) for each B.
Proof. It suffices to show that A � I. Suppose firstly that the
operator U in thepolar decomposition A = UT of A is an isometry.
The operator AU∗ = UTU∗
is non-negative and non-compact. By Lemma 8.9, AU∗ � I. Since
U∗H = H, byLemma 8.7 (ii), we have A � I.
If U is a coisometry, then U∗A = T is a positive non-compact
operator. ByLemma 8.9, T � I, and since ker(U∗) = 0, by Lemma 8.7
(ii), we get A � I. �
9. A-expanding operators
In operator theory, especially in dealing with interpolation
problems, one oftenneeds to consider Hilbert (or Banach) spaces
with two norms and study operatorswith special properties with
respect to these norms. The main purpose of thissection is to show
that Kolmogorov n-widths can be used to describe WOT-closuresof
some sets of operators given by conditions of this kind. Our
interest to suchconditions is inspired by the theory of linear
fractional relations, see [11] and [12].
Let X be a Banach space and A ∈ L(X ) be a compact operator with
aninfinite-dimensional range. It determines a semi-norm ‖x‖A = ‖Ax‖
on X . Weconsider the set E(A) of all operators R that increase
this semi-norm: ‖Rx‖A ≥‖x‖A for each x ∈ X , that is,
E(A) := {R ∈ L(X ) : ||ARx|| ≥ ||Ax|| ∀x ∈ X}. (19)It turns out
that the problem of description of E(A) is a dual version of
the problem considered in previous sections: the following dual
characterization ofE(A) relates it with covering operators.
Lemma 9.1. Let a Banach space X be reflexive. An operator R ∈
L(X ) satisfiesR ∈ E(A) if and only if R∗ ∈ G(K), where K =
A∗(BX∗).
Proof. Assume that R∗ ∈ G(K), that is, R∗K ⊃ K. Then||ARx|| =
sup
f∈BX∗|f(ARx)| = sup
f∈BX∗|(R∗A∗f)(x)| ≥ sup
f∈BX∗|(A∗f)(x)| = ||Ax||,
(20)for each x ∈ X. Thus R ∈ E(A).
Conversely, if R∗ /∈ G(K), then there is f ∈ K \ R∗K. The set
R∗K isweakly closed. By the Hahn–Banach theorem and reflexivity of
X there is x ∈ Xwith |f(x)| > supg∈R∗K |g(x)| = ‖ARx‖. Since
|f(x)| ≤ ‖Ax‖ we obtain thatR /∈ E(A). �
We denote the WOT-closure of E(A) by WE(A).
Corollary 9.2. Let X be a reflexive Banach space, A an operator
on X . Then{R∗ : R ∈ WE(A)} = WG(K), where K = A∗(BX∗).
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Operator ranges 21
Proof. Since X is reflexive the map R→ R∗ from L(X ) to L(X ∗)
is bicontinuousin the WOT-topologies. Hence the result follows from
Lemma 9.1. �
Corollary 9.3. Let X be reflexive. If A ∈ L(X ) is such that the
sequence{dn(A∗(BX∗))}∞n=0
is lacunary, then WE(A) is contained in the set of all operators
for which kerA isan invariant subspace.
Proof. Follows immediately from Lemma 3.5 if we take into
account the observa-tion that kerA is an invariant subspace of R if
and only if A∗X ∗ is an invariantsubspace of R∗ (that is, if and
only if R∗ ∈ AK). �
Applying Theorem 2.1, we obtain the converse inclusion:
Corollary 9.4. The set of all operators preserving kerA is
contained in WE(A). IfkerA = {0}, then WE(A) = L(X ).
Applying Theorem 4.6, we get
Corollary 9.5. If X is a separable Hilbert space and A ∈ L(X )
is such that thesequence of s-numbers of A is not lacunary, then
WE(A) = L(X ).
We can summarize Hilbert-space-case results in the following
way:
Theorem 9.6 (A complete classification in the Hilbert space
case). Let X be aseparable Hilbert space.(i) If the sequence of
s-numbers of A is not lacunary, then WE(A) = L(X ).(ii) If the
sequence of s-numbers of A is lacunary, then WE(A) coincides with
theset of operators for which kerA is an invariant subspace.
Finally, using Theorem 4.10 we obtain a result on the ultra-weak
closure ofWE(A):
Corollary 9.7. Let A ∈ L(H) be such that its sequence of
s-numbers is not lacunary.Then the closure of the set (19) in the
ultra-weak topology coincides with L(H).
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Acknowledgment
The authors would like to thank Heydar Radjavi for a helpful
discussion and hisinterest in our work.
M. I. OstrovskiiDepartment of Mathematics and Computer
ScienceSt. John’s University8000 Utopia ParkwayQueens, NY
11439USA
e-mail: [email protected]
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Operator ranges 23
V. S. ShulmanDepartment of MathematicsVologda State Technical
University15 Lenina streetVologda 160000RUSSIAe-mail: shulman
[email protected]